I don't have the tune handy but I do have a busking book that gives the usual key for the song as Eb and the starting note as Bb. That's F for the tenor and the starting note C. Have you tried sticking your knee into the bell to get that A? Putting the song into a comfortable key is the best solution.
I play a low Bb baritone and run across lots of low As. It's a pain. Sometimes I take just the A up the octave or find an alternative note that fits.
I disagree to some extent with transposing this one as a matter of course. There is a value to knowing tunes in the standard key (and being able to play them in others, of course). Except for that one figure, Eb is a good key for this tune on tenor. What I usually do is change the figure. Instead of A-Bb-D play something like C-C#-D.